OK, I think I found what's causing this, but I'm not quite sure how I'm going to remedy it just yet. On every three light section of flex strip that is not an end point (indicated by a black arrow going right), the DOUT pin of the 1809 chip goes to one of the two small traces on the back of the strip (the one nearest the V+ Pad side of the strip). That trace goes across the cutting point (where the two sets of pads are) to the next set of three lights. It then comes back to the left (assuming the input side is on the left) and feeds the DOUT pad AND the DIN Pad. See attached pictures for more details. Seems pretty silly if you ask me.
I came up with two possible solutions, but I'm not sure which is best, perhaps someone has a better idea
1) Scrape away the trace on the backside of the strip and get my DOUT from there
2) Try to solder directly to the DOUT pin on the chip.
I was trying to come up with a more creative way to keep the trace and pads intact, but every way I try, I either lose the ability to get data out, or lose the ability to put data in on the next set of strips.